So, this took me a little while to compose…
I have read the very nice work of Alain again, who is calculating the anchor load - also in the presence of dynamic loads - in the force-time domain:
alain.fraysse.free.fr/sail/rode/forces/forces.htm
He models a gust as a stronger wind force that gets switched on at a certain point in time and then stays on “forever”.
He starts by discussing the linear case (like a rope), where the gust leads to an overshooting over the position the boat would take had the gust been a static wind. Because he discusses the linear case, the point of maximal overshooting is exactly a mirror point of the initial static point, where the position where gust and rope tension balance out is exactly in the middle.
The first diagram shows what happens. It is a plot of the displacement of the boat (away from the anchor) as a function of load. So, when I integrate underneath the area (force times distance), I get directly the energy stored in the elastic rope. It is simply the area when looking at the graph sideways.
The areas in light red are energies that are not (yet) loaded into the rope (the dark red area), but rather go into the kinetic energy of the boat. At the yellow point the pull back force of the rope equals the gust load. But as the boat has some velocity, it overshoots until it has reached the red point, the most extreme tension in the rope. The two areas labelled “E” must be the same, if no energy is lost somewhere.
Please note the loads at the green, yellow and red points. The difference between green and yellow is the same as the difference between yellow and red. This is the best possible scenario and leads to the smallest possible maximal overshooting tension.
Nonlinearity makes the result worse. The next diagram is the same as before, but now not with an ideal spring, but rather a chain acting as a spring. Again, the gust keeps blowing even when the boat has reached the extreme position with the highest displacement away from anchor. But now the distance between the yellow and the red point is much larger than the distance between the yellow and the green point. In fact, the most extreme point, the red point, now has about 4 times the load of the static wind. That is a lot!
There are two graphs in this diagram acually. One for deep water and one for shallow water. There is no big difference between the two. And in this scenario the vessel’s mass does not matter at all.
Finally, I show the same graphs, but now assuming that the gust only blows for a few seconds, resulting in this “energy spike” to the right. So, in this case, the vessel is still moving backwards when the gust is stopping allready.
Overall, the red points are much more to the left than in the previous graphs, which is obvious, as the total energy transferred to the boat is less. But one can also see now that in shallow water the red point is much further to the right than in deep water.
That was my initial point when I started this thread. Shallow water can be dangerous in the presence of gusts. But a very elastic rope can help to address this shortcoming of the chain.
But, as we see now, it depends on the nature of the gust. When it is simply a switching on of a higher wind force, then it does not matter whether it is shallow or not. But when it is a sudden and short burst of wind, then it does matter.
The reality is - as so often - somewhere inbetwen…
BTW - This last graph also explains why large commerical vessels do not see this effect at all. Because of their large mass m, according to F = m * a, their acceleration in a gust will be much smaller, and hence the displacement backwards away from the anchor is also so much less. This results in a very small energy transfer E, and so a very tiny overshooting.
Older pleasure crafts that are rather heavy for their windage area also are less affected by these dynamic loads - precisely for the same reason. They hardly move at all in a gust.
Cheers, Mathias